Motion-Enabled Tomography via Gaussian Mixture Models

Can Evren Yarman; Daniel Burrows; Ozan \"Oktem

arxiv: 2605.18210 · v1 · pith:Q5SSOBIInew · submitted 2026-05-18 · 🧮 math.NA · cs.NA

Motion-Enabled Tomography via Gaussian Mixture Models

Daniel Burrows , Can Evren Yarman , Ozan \"Oktem This is my paper

Pith reviewed 2026-05-20 00:16 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords tomographyGaussian mixture modelsmotion-enabled reconstructionray transforminverse problemsparametric modelingspatiotemporal tomography

0 comments

The pith

Gaussian mixture models with independent per-component motions enable reconstruction of moving objects in tomography by decoupling into two subproblems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a method for tomographic reconstruction of objects in motion by modeling them as Gaussian mixture models. Each Gaussian in the mixture is given its own angular velocity, projectile motion, and geometric parameters. The method takes advantage of the closed-form expression for the ray transform of a Gaussian to make predictions and compute gradients efficiently in data space. The reconstruction is split into two separate inverse problems that are solved by minimizing specific objective functions. The approach is designed to work for objects in any number of dimensions as long as the motion provides adequate views and the object fits the mixture approximation.

Core claim

By representing the object as a collection of Gaussians each with independent motion parameters and using the closed form of the ray transform, the spatiotemporal tomography problem decouples into two sub-inverse problems. Solutions are then the minimizers of derived task-specific objective functions, resulting in an algorithm that applies to Euclidean spaces of arbitrary dimension and was demonstrated on a 2D case with five Gaussians on intersecting trajectories.

What carries the argument

The parametric spatiotemporal Gaussian mixture model with per-component angular velocity and projectile motion parameters, whose ray transform has a closed-form expression that enables the decoupling of the inverse problem.

If this is right

The resulting algorithm applies to objects in Euclidean space of arbitrary dimension.
Accurate reconstruction is achieved for a 5-Gaussian GMM with intersecting trajectories in 2D simulation.
The method provides a foundation for further work with noisy data, 3D objects, and non-rigid body dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This technique may apply to medical imaging of moving tissues or industrial monitoring of rotating parts.
Extensions could test the method on experimental data rather than simulations to assess robustness.
Similar parametric approaches might address other inverse problems involving motion and projections.

Load-bearing premise

The relative motion between the object and the sensing apparatus provides sufficient angular coverage, and the object can be accurately approximated by a GMM with each component having independent angular velocity and projectile motion parameters.

What would settle it

Running the algorithm on a simulated dataset where the object cannot be well approximated by a GMM or where angular coverage is insufficient, and observing whether the recovered parameters deviate significantly from the known ground truth.

Figures

Figures reproduced from arXiv: 2605.18210 by Can Evren Yarman, Daniel Burrows, Ozan \"Oktem.

**Figure 8.1.** Figure 8.1: Acquisition geometry, with one source located at ( [PITH_FULL_IMAGE:figures/full_fig_p017_8_1.png] view at source ↗

**Figure 8.2.** Figure 8.2: Projection data corresponding to N = 5 Gaussian particles in projectile motion. where E ∈ R 2×2 is strictly upper triangular with entries given by i.i.d. samples from N (0, 1). • Attenuation: αn = 12.5 for all n; (7.3) replaces these with least squares with enforced non-negativity estimates. • Angular velocity: Θn = 0 for all n. The Stage 2 initialization replaces these via a grid search over a specified… view at source ↗

**Figure 8.3.** Figure 8.3: Projection data and corresponding color coded estimated modes at times [PITH_FULL_IMAGE:figures/full_fig_p019_8_3.png] view at source ↗

**Figure 8.4.** Figure 8.4: Left: The detected modes and the mode maps [PITH_FULL_IMAGE:figures/full_fig_p019_8_4.png] view at source ↗

**Figure 8.5.** Figure 8.5: Left: The simulated particle states sampled at times [PITH_FULL_IMAGE:figures/full_fig_p021_8_5.png] view at source ↗

**Figure 8.6.** Figure 8.6: The same visualization scheme as in Figure [PITH_FULL_IMAGE:figures/full_fig_p022_8_6.png] view at source ↗

**Figure 8.7.** Figure 8.7: First row: the unknown, simulated particles. Second row: the Stage 1 initial [PITH_FULL_IMAGE:figures/full_fig_p023_8_7.png] view at source ↗

read the original abstract

Recovering physical properties of objects in motion is a core task across scientific and industrial applications. When the relative motion between the object and the sensing apparatus provides sufficient angular coverage, Computerized Tomography offers a powerful means of reconstruction. For such scenarios, we propose a parametric spatiotemporal model applied to Gaussian Mixture Models (GMM), in which each constituent Gaussian is parameterized by its own angular velocity, projectile motion, and geometry. GMM are a suitable means of reconstruction because they (i) admit accurate approximations in object space and (ii) have a closed form expression under the ray transform; enabling efficient forward predictions and exact gradient computations in data space. By decoupling the reconstruction problem into two sub-inverse problems, we characterize solutions as minimizers of task-specific objective functions that are derived and solved by utilizing the properties of (ii). The resulting algorithm we provide is applicable to objects in Euclidean space of arbitrary dimension. We validate the method on a simulated 2D problem, achieving accurate reconstruction of a 5-Gaussian GMM with intersecting trajectories. This also provides a foundation for further experimentation in settings with noisy data, 3D objects, and non-rigid body dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a GMM with per-Gaussian motion params for tomography and splits reconstruction into two subproblems, but the split may miss joint critical points.

read the letter

Hi colleague, The punchline on this arXiv paper is that it gives a parametric way to reconstruct moving objects in tomography by representing them as Gaussian mixtures where each Gaussian has its own angular velocity and projectile motion parameters. They then decouple the reconstruction into two sub-inverse problems to make it solvable. What the paper does well is exploit the fact that GMMs have a closed-form expression under the ray transform. This means they can compute the forward model and gradients exactly and efficiently without discretization issues in the object space. The 2D simulation with a 5-Gaussian model and intersecting trajectories reconstructs both the shapes and the motions accurately enough to demonstrate the concept. Framing it for arbitrary Euclidean dimensions is also a solid point. On the soft spots, the decoupling is the part that stands out as needing more attention. The stress-test concern is on point here: the positions of the Gaussians at each projection time are nonlinear functions of the motion parameters, so the data term links them all together. Characterizing solutions as separate minimizers of task-specific objectives might not correspond to a stationary point of the original joint functional. The paper would be stronger with an explicit check that their algorithm reaches the same place as joint optimization or a proof that the split preserves the critical points. The validation is another area that is light. It's a noise-free 2D case with known number of components, and while it works, adding noisy data tests or 3D examples would help show robustness. Choosing the number of Gaussians is left as a free parameter, which is typical but practical issues around that could be discussed. This paper is for researchers in mathematical imaging and inverse problems who are looking at dynamic tomography scenarios in science or industry. A reader who wants to see how parametric models can handle motion without going full nonparametric would get value from the details on the forward model and the splitting strategy. It deserves a serious referee because the core math is built on standard GMM and transform properties, and the simulation provides initial evidence that it can work. I would recommend sending it for peer review, with the expectation that reviewers will probe the optimization equivalence and ask for more extensive validation.

Referee Report

1 major / 2 minor

Summary. The paper proposes a parametric spatiotemporal model for tomography of moving objects based on Gaussian Mixture Models (GMMs), with each Gaussian component having independent parameters for angular velocity, projectile motion, and geometry. It decouples the inverse problem into two sub-problems whose solutions are characterized as minimizers of derived task-specific objective functions that exploit the closed-form ray transform of the moving GMM. The resulting algorithm is asserted to apply in Euclidean spaces of arbitrary dimension and is validated via a 2D simulation recovering a 5-Gaussian GMM with intersecting trajectories.

Significance. If the decoupling is shown to preserve critical points of the joint objective, the approach would provide an efficient, gradient-exact method for motion-enabled tomography that scales to higher dimensions by leveraging standard GMM properties under the ray transform. The 2D simulation offers a concrete demonstration, though quantitative error metrics are not detailed in the abstract.

major comments (1)

Abstract: the central claim that decoupling the reconstruction into two sub-inverse problems yields independent minimizers of task-specific objectives is load-bearing for the method. Because each Gaussian's position at projection time depends nonlinearly on its own motion parameters, the data-fidelity term couples geometry and motion variables; the manuscript provides no explicit verification that the separate minimizers coincide with a stationary point of the original joint functional or that alternating minimization converges to the joint reconstruction.

minor comments (2)

Abstract: the validation statement 'achieving accurate reconstruction' is not accompanied by quantitative error metrics, noise levels, number of projections, or comparison to a joint-optimization baseline.
Abstract: the claim of applicability to arbitrary dimensions is stated but the only reported experiment is 2D; a brief outline of the 3D extension or scaling argument would clarify the generality.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We address the major comment below and will revise the paper to strengthen the theoretical justification of the decoupling approach.

read point-by-point responses

Referee: Abstract: the central claim that decoupling the reconstruction into two sub-inverse problems yields independent minimizers of task-specific objectives is load-bearing for the method. Because each Gaussian's position at projection time depends nonlinearly on its own motion parameters, the data-fidelity term couples geometry and motion variables; the manuscript provides no explicit verification that the separate minimizers coincide with a stationary point of the original joint functional or that alternating minimization converges to the joint reconstruction.

Authors: We thank the referee for highlighting this important point. In the manuscript we separate the variables into per-Gaussian geometric parameters and motion parameters, then exploit the linearity of the ray transform together with the closed-form expression for each moving Gaussian to obtain two task-specific objective functions that are minimized independently. The referee is correct that the current version does not contain an explicit proof that the resulting minimizers are stationary points of the joint functional, nor does it discuss convergence properties of any iterative scheme. Our procedure solves the two sub-problems sequentially rather than by alternating minimization; we will clarify this distinction in the revision. To address the concern directly, we will add a short subsection deriving the stationarity condition of the joint objective and showing that, under the independent per-component parameterization and sufficient angular coverage, setting the separate gradients to zero recovers the same critical points. This addition will be placed in the Methods section immediately after the derivation of the decoupled objectives. revision: yes

Circularity Check

0 steps flagged

No circularity: decoupling uses independent closed-form forward model

full rationale

The derivation characterizes solutions via minimizers of objectives built from the closed-form ray transform of each moving Gaussian component. This property of GMMs under the ray transform is a standard mathematical fact independent of the target reconstruction or any fitted parameters in the present work. The split into two sub-inverse problems is introduced as an algorithmic choice to exploit that closed form for separate geometry and motion estimation; it does not rename a fitted quantity as a prediction, invoke a self-citation uniqueness theorem, or smuggle an ansatz. Validation on simulated intersecting-trajectory data supplies an external check. The central claim therefore remains self-contained against the forward model rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard mathematical properties of Gaussians under the ray transform and the modeling assumption that GMMs provide accurate object approximations; the number of components is a free modeling choice.

free parameters (1)

number of Gaussians
Selected as 5 for the 2D simulation validation; the general method allows arbitrary numbers as a modeling decision.

axioms (2)

domain assumption GMM admit accurate approximations in object space
Invoked in the abstract as a key reason GMMs are suitable for reconstruction.
standard math Gaussians have a closed form expression under the ray transform
Stated in the abstract as enabling efficient forward predictions and exact gradients.

pith-pipeline@v0.9.0 · 5733 in / 1276 out tokens · 79582 ms · 2026-05-20T00:16:27.331409+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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